The characteristic behavior of a fluid is dependent on the Reynolds number of the flow.
A fluid flow stays laminar between Reynolds numbers of 1000 and a million. This range however,
is just a guideline and the exact range depends on various factors such as free-stream conditions,
surface roughness and others. One question you might have is, what happens to the flow
beyond this range? The short answer is that it transitions to a turbulent flow! Let us now
spend some time understanding what this means and what happens as we exceed the laminar flow range.
A laminar flow is characterized by smooth organized motion of the fluid layers.
However, as the Reynolds number of a laminar flow increases, the orderly nature is lost and
a chaotic flow pattern, referred to as a turbulent flow is established. The process through which a
laminar flow becomes turbulent consists of multiples stages and is called transition,
or onset of turbulence. For most industrial fluid flows, the Reynolds number at which the transition
occurs is relatively small, meaning that these flows are almost always turbulent.
Osborne Reynolds demonstrated the laminar – turbulent transition process
through his famous experiment where he visualized the motion of water flow at different flow rates,
by introducing dyed water into it. At low flow rates, he noticed that the stream of dyed water
remained distinct throughout the length of the tube. However, as the flow rate is increased
beyond a certain value, the dye diffused across the entire cross-section of the tube. In fact
it is through these experiments that he proposed a similarity criterion, which eventually became
the Reynolds number, relating the inertial and viscous forces in a fluid flow. The phenomenon of
transition is dependent on many factors such as surface roughness, free stream turbulence level,
etc. Also, the type of transition is different based on the flow situation.
For boundary layer flows such as flow over a solid surface, the primary instability/transition
mechanism is through the amplification of Tollmien-Schlichting waves. These waves are
two dimensional disturbances that travel in the mean flow direction, which eventually get
amplified and force the uniform boundary layer to transition to the three-dimensional turbulent.
Based on numerous theoretical and experimental studies,
the location of the transition location for a flat plate boundary layer was predicted to be at
an axial distance on the plate where the Reynolds number is around 0.35 to 1 million. For free-shear
layer flows, where fluids are moving at different velocities relative to each other, the instability
mechanism that describes the transition process is referred to as the Kelvin-Helmholtz instability.
This type of instability visually manifests in the form of vortical structures or ocean-like waves,
often seen in cloud patterns. The most famous example of KH instability is Jupiter's Red Spot.
I know what you might be thinking now. What causes transition?
In order to understand that, we need to study the concept of stability.
Have you ever played with a hit me doll? No matter how hard you punch it, it always comes back
to the upright position after a few oscillations. That is, it withstands disturbances and returns
to its original, undisturbed state. Such systems are considered to be stable systems.
Let us consider the classic example of a ball in a bowl to demonstrate different states
of a system. If the bowl is right side up and the ball is set to roll,
it oscillates for some time but eventually arrives at an equilibrium point. This is a stable system.
Similar to the hitme doll example. If we instead, invert the bowl and place the ball
at the top of the bowl and give it a small push, the ball will roll away, all the while
increasing its speed, and will never come back to the original state. This is an unstable system.
There can be other states such as neutrally stable, I.e. when a ball is placed on a
flat surface. If we now push the ball, it moves for a certain and remains stable at a
different location, and does not come back to its original state. From fluid dynamics point of view,
for a fluid flow to be stable, it needs to be able to absorb and exhibit no noticeable
changes when external disturbances such as flow property variations, surface roughness
changes etc, are encountered. In unstable fluid flows, any small disturbance to the flow results
in visually noticeable flow changes, which eventually leads to laminar-turbulent transition.
So, how do we predict the stability behavior of fluid flows? Well, today, we can perform numerical
simulations to identify the conditions at which the fluid flow transitions.
However, a few hundred years ago, scientists didn't have this luxury. In a few cases,
they could do experiments but for most situations, they had to resort to theoretical predictions.
Most approaches used to predict stability of fluid flows
fall under the category of small disturbance theory. In any typical small disturbance analysis
The basic flow variable is disturbed using a small perturbation.
This new variable is substituted into the governing equations of the particular fluid flow.
From the new set of equations, equations governing the evolution of the perturbation,
known as the disturbance or the perturbation equations are derived
These equations are then linearized to remove the higher
order terms. The resultant equation should be homogenous
This homogenous equation is then solved and the eigen values are calculated,
which are examined to determine the stability of the fluid flow.
Let us implement this process in predicting the instability character
for free-shear flows I.e. the Kelvin-Helmholtz instability.
Consider two flows of different velocities and densities separated by a horizontal interface.
The flows are assumed to be potential I.e irrotational, incompressible and inviscid.
For such flows, the velocity potential and the hydrostatic pressure distribution is
given by the following equations. Since the flows are potential, they satisfy
the laplace equation. Following our established procedure for small disturbance analysis,
let us impose a perturbation of the following form for both the velocity potential equations.
Let us impose that the disturbances are finite only at the interface and can be neglected
far away from the interface. This assumption is basically our far-field
boundary condition. It is easy to show that the disturbances also satisfy the laplace equation.
Due to the perturbation, the interface is disturbed to the form shown here.
There are two types of interface boundary conditions this system needs to satisfy.
Kinematic and Dynamic. The kinematic condition at the interface requires that vertical
velocity of the two fluids match the interface. Mathematically, this can be written as shown here.
Since the fluid flows are assumed to be potential, the pressure in each of the fluids satisfies
the unsteady Bernoulli's equation, shown here. For the base undisturbed flow, this equation reduces
to the form given here. The Dynamic condition at the interface requires that the pressure is
constant across the interface. Based on this, we can write the following relations at the interface
between the two fluids, both before and after imposing the disturbance.
If we assume that the perturbations are weak and only result in very small displacements
of the interface, and also ignore the higher ordered terms, the relations we
obtained from both the kinematic and the dynamic conditions, can be recast in form shown here.
We now have these three equations, the disturbance laplace equation and the far-field condition, that
the perturbations need to satisfy. Let us assume simple two dimensional disturbances as shown
here, where alpha is a real number and represents the wavenumber and omega is a complex number
represents the frequency of the perturbation. The disturbance will be amplified and make
the fluid interface unstable if the imaginary part of omega is positive. If we substitute the
perturbation in the laplace equation and employ the far-field boundary condition,
we will obtain the amplitudes of the perturbation as shown here. Using these amplitudes and the
interface kinematic condition, we can extract the values of the unknown quantities as shown here.
Substituting the perturbations into the dynamic condition and extracting
the roots of the subsequent quadratic equation, we will obtain the eigen values.
The stability of the fluid system is dependent on the value of the highlighted term.
The disturbance is unstable if the value is negative, neutral if the value is 0 and stable
if the value is positive. Therefore, the unstable condition is given by the relation shown here.
Till now we discussed the prediction of instability
for free-shear flows. A similar methodology can be followed to characterize the stability
for parallel viscous flows, which describe the flow inside laminar boundary layers.
Let us directly start with the perturbation form of the 2D incompressible Navier stokes equations.
If we assume that the disturbance is in the form of a wave travelling in the x-direction,
the stream function of a single mode in the perturbation can be written in the following form.
Here w is a complex number where the real part represents the frequency of the mode and the
imaginary part is the amplification factor. If wi<0, the wave is damped and therefore the flow
is stable. However, if wi>0, instabilities are present in the system, that eventually
get amplified. From the stream function, we can obtain the velocity components as shown here.
If we now substitute these velocity components
into the disturbance equations and rearrange them such that the pressure term is eliminated,
we obtain a 4th order ordinary differential equation for the amplitude function.
This approach was independently developed by William Orr and Arnold Sommerfeld,
and this 4th order equation which describes the hydrodynamic stability characteristic
of viscous parallel flows is referred to as the Orr-Sommerfeld equation.
This equation represents an eigen value problem with the following boundary conditions.
The results obtained from solving the equation can be visualized in the form of curves
that are referred to the as the neutral stability curves or more commonly as thumb curves.
The region inside the thumb represent conditions where the disturbances are amplified.
The lower limit of the Reynolds number at which the disturbances start amplifying
is called the critical Reynolds number. For a Blasius boundary layer profile,
the critical Reynolds number is 91000 based on the axial distance from the leading edge of the plate.
The corresponding wave parameters are shown here. The smallest unstable wavelength is around
6 times the thickness of the boundary layer. This implies that the Tollmein Schlichting waves,
which are natural disturbances in the flow, are long compared to the boundary layer.
The maximum phase velocity of these waves
is around 0.4 which indicates that the TS waves travel slowly and arise near the wall.
Finally, the transition to turbulence occurs at around a Reynolds number of 3 million,
which is around 30 times further downstream compared to the location of initial instability.
The process of emergence of the TS waves
and the eventual transition to turbulence for a flow over a smooth surface follows these steps.
At the leading edge of the plate, there is stable laminar flow until a location
where the Reynolds number if the flow is below the critical reynolds number.
Beyond this point, 2D TS waves emerge and start travelling downstream.
These 2D waves quickly become unstable and take the shape of three dimensional hairpin eddies.
These vortices then breakdown creating a cascade of turbulent eddies,
together forming turbulent spots.
These spots then combine to form a fully turbulent region thereby completing the transition process.
That brings us to the end of our discussion on stability
characteristics of laminar flows and the process of transition to turbulent flow.